Idea

Definition

We define the Reidemeister moves in terms of link diagrams.

The conventions are that part of the diagram is shown with the ‘move’ indicated, but that outside that, the diagram of the link is unchanged and that no other part of the diagram appears at the crossings that are changed by that move.

Note

In manipulating a link diagram, the changes obtained by elongating part of the diagram, rotating it, deforming the plane, etc., are not counted as changing the structure of the diagram provided they do not change any crossing, and so they may be done without comment. The ‘moves’ outlined here do change things locally near crossings.

First Reidemeister moves (R1)

The local configurations

are interchangable;

Second Reidemeister moves (R2)

The configurations

are interchangeable;

Third Reidemeister move (R3)

The configurations

are interchangeable.

Often the terminology is used that two link diagrams, DD and D′D' are isotopic or isotopic by moves if DD can be transformed into D′D' by some sequence of the three Reidemeister moves, R1, R2 and R3. They are called regularly isotopic if the transformation can be done without using R1. It is clear that both isotopy and regular isotopy give equivalence relations on link diagrams.

The following theorem is a form of Reidemeister’s theorem:

Properties

Reidemeister’s Theorem

Theorem

Two knot diagrams correspond to isotopic knots precisely if they can be connected by a sequence of applications of the three Reidemeister moves.

Because of this, we can hope to use the Reidemeister moves to verify invariance of a potential link invariant. For instance, as none of the moves alters the number of components of a link diagram, the number of components is an isotopy invariant (which is not at all surprising!) We can conclude that the Hopf link and the Borromean rings are not isotopic.

A deeper observation is that the linking number is a link invariant, for which see that entry.

3-colourability is a knot invariant as is easy to check. This shows, for instance, that the trefoil knot is not equivalent to the unknot, and hence shows, very simply, that non-trivial knots exist. It also shows that the figure eight knot is not equivalent to the trefoil.

(Links to other instances of this sort of argument will get put here later on.)